PreprintCC BY-NC-ND 4.0HAL
Coupling-convexity of a diffusion with respect to its starting position
Abstract
In the present paper, we consider a d-dimensional time-homogeneous stochastic differential equation with diffusion matrix a and no drift coefficient. We check that the convexity in the starting position x of the distribution of its solution on the path space when the codomain is endowed with the convex order is equivalent to the convexity of Rd∋ x↦ √(ζ*a(x)ζ) for each ζ∈Rd. To do so, we introduce the seemingly stronger notion of coupling-convexity in the starting position and check that it is also equivalent to the convexity of the above square-rooted functions.
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BibTeX
@misc{jourdain2026couplingconvexity,
title = {{Coupling-convexity of a diffusion with respect to its starting position}},
author = {Benjamin Jourdain},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11528},
eprint = {2610.11528},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788093v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Coupling-convexity of a diffusion with respect to its starting position
AU - Benjamin Jourdain
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.11528
UR - https://hal.science/hal-05788093v1
AB - In the present paper, we consider a d-dimensional time-homogeneous stochastic differential equation with diffusion matrix a and no drift coefficient. We check that the convexity in the starting position x of the distribution of its solution on the path space when the codomain is endowed with the convex order is equivalent to the convexity of R^d∋ x↦ √(ζ^*a(x)ζ) for each ζ∈R^d. To do so, we introduce the seemingly stronger notion of coupling-convexity in the starting position and check that it is also equivalent to the convexity of the above square-rooted functions.
ER -